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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>, <font color="Maroon">X2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:148"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font>
 ;<br/>

<b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X1</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X2</font>,<font color="Maroon">Y</font>;<br/>



<b>thus </b><a NAME="E2:148"/>
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font> implies <font color="Maroon">f1</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_1"><b>proof </b></a><div class="add">

<a NAME="E1:148_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, <a class="ref" href="tsep_1.html#T30">TSEP_1:30</a></i>;<br/>
<a NAME="E2:148_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<b>assume </b><a NAME="E3:148_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font> )
 ;<br/>

<a NAME="E4:148_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f1</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#T33" target="_self">Th33</a>, <a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E5:148_1"/>
<font color="Maroon">f1</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:148"/>
( <font color="Maroon">f1</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span> implies ( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tmap_1.html#K10" target="_self">union</a> <font color="Maroon">f2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font> ) )
 <b>by </b><i><a class="ref" href="tmap_1.html#T27" target="_self">Th27</a></i>;<br/>


</div>
